NOTE: This record of what has been posted to PEIRCE-L
has been nodified by omission of redundant quotations in
the messages. both for legibility and to save space.
-- Joseph Ransdell, PEIRCE-L manager/owner]

Here follows some comments and analysis of the Peirce quote helpfully
supplied by Gary Richmond. Generally I aim to keep an eye on the thesis of
the non-reducibility of triadic relations.

Peirce writes:
----quote-------------------
CP 1.345. I will sketch a proof that the idea of meaning is irreducible to
those of quality and reaction. It depends on two main premisses. The first is
that every genuine triadic relation involves meaning, as meaning is obviously
a triadic relation.
----pause----------

Notice here that Peirce speaks of "every genuine triadic relation," implying
that what appears triadic may yet not be. Every genuine triadic relation
"involves meaning."
Some apparently triadic relations, may, it seems be properly analyzed into
some-thing less than triadic. Intuitively, I follow Peirce to the extend of
thinking that our talk of meaning involves the reference of signs to objects
in light of an interpretation.
But is this intuitive notion of meaning "reducible" to component or
non-triadic relations, say, the relation of sign to object, the relation of
sign to interpretation and the relation of object to interpretation? That
meaning is not reducible to quality and reaction seems, so far, a perhaps
less interesting claim.

A second premise:

----Peirce continued-----
The second is that a triadic relation is inexpressible by means of dyadic
relations
alone.
---pause----------------------

The second premise of this argument for Peirce's seems to be the
irreducibility of triadic relations to dyadic relations. So, if I read this
argument correctly then Peirce is using the assumption of the
non-reducibility of triadic relations to dyadic relations, here explicitly
taken as a premise, in order to argue for a a distinct claim -- "that the
idea of meaning is irreducible to those of quality and reaction." This looks
like an argument for the irreducibility of the category of thirdness based on
the assumption of the irreducibility of triadic realtions. So, it seems we
are not about to find an argument for the irreducibility of triadic
relations.

----Peirce continued---------
Considerable reflexion may be required to convince yourself of the first of
these premisses, that every triadic relation involves meaning. There will be
two lines of inquiry. First, all physical forces appear to subsist between
pairs of particles. This was assumed by Helmholtz in his original paper, On
the Conservation of Forces.+1
Take any fact in physics of the triadic kind, by which I mean a fact which
can only be defined by simultaneous reference to three things, and you will
find there is ample evidence that it never was produced by the action of
forces on mere dyadic conditions. Thus, your right hand is that hand which is
toward the east, when you face the north with your head toward the zenith.
Three things, east, west, and up,
are required to define the difference between right and left. Consequently
chemists find that those substances which rotate the plane of polarization to
the right or left can only be produced from such [similar] active substances.
They are all of such complex constitution that they cannot have existed when
the earth was very
hot, and how the first one was produced is a puzzle. It cannot have been by
the action of brute forces. For the second branch of the inquiry, you must
train yourself to the analysis of relations, beginning with such as are very
markedly triadic,
----end quote-------

In this last passage, a triadic relation is defined as one which requires
reference to three things. That strikes me as somewhat problematic. Are there
facts which can only be defined by reference to three things? Well, perhaps;
and perhaps meaning is one of them. But in any case, I wonder how we are to
take the intuitive use of the phrase "three things." What is taken as one
thing for one purpose may be taken as more than one for some other purpose,
or so it seems. So, I ask, whether the reference to three things, on which
the notion of triadic facts and relations appears to depend, is something so
firm as to lead us to generally deny that what are taken as three things for
some purpose may be taken as more or less than three for some other purposes?
Twelve eggs are also 1 dozen. Right? Or is this somehow just common-sense
nominalism, which we want to get beyond in a critical spirit?

Thus we see the origin and meaning of the term "numerical identity".
For the account that I gave last time, enumerating the first three
generations of relations over the universe X = {i, j, k} will ever
after serve to remind us of one of the things that can give us the
most confidence that we have comprehended the numerical identities
of any types of entities, to wit, a putative ability to count them.

Let us make a few observations of general bearing that we can
already see exhibited in this early but blossoming universe X,
and also take the occasion to set down a few bits of notation.

Let me introduce a bit of language that comes up here.
Very roughly speaking -- for speaking this way ignores
a point of subtlety concerning the distinction between
extensions and intensions, and another point concerned
with the distinction between the "relative term" and
the "relation" proper -- Peirce called the elements
of a relation, its tuples, by the suggestive name
of "elementary relations". Let us do likewise.

For example, the elementary 2-adic relations that
serve as a basis for all of the 2-adic relations
over X are just the 3 x 3 = 9 ordered 2-tuples
that I list here:

i:i, i:j, i:k,
j:i, j:j, j:k,
k:i, k:j, k:k.

I hope that you will discover this form to be
a suggestive array and not a block to inquiry.

Now that we have an initial notion of what a relation is,
namely, an aggregate, class, collection, set, logical sum,
by whatever name of a similar sort you may wish to call it,
of ordered tuples, and now that we will forever after never
confuse a relation with one of its elemental tuples -- for,
yes, indeed, even a set that consists of a single element,
a "singleton" so-called, is counted as a different entity
from the element thereof -- we may begin to consider the
types of operations to which these relations are subject.

"Genuine" is a term of art in Peirce's philosophy, and is not equivalent to
"real" as opposed to "apparent". I am distributing separately a
collection of passages from the CP compiled some time ago. Bear in mind
that they were acquired by a string search, and I have only minimally edited
them for legibility. The first paragraph is the same as the one Gary
quoted, but contains a little more, I believe..

(Caveat lector! Note that omitted intervening paragraphs may sometimes
provide important context which is missing here since the paragraphs quoted
here were identified and compiled by use of a string search. The search was
for uses of _degenerate_ rather than _genuine_, but his technical usage of
the latter is made clear in several of the passages. -- JOSEPH RANSDELL)

Collected Papers 1.345
345. I will sketch a proof that the idea of meaning is irreducible to those
of quality and reaction. It depends on two main premisses. The first is that
every genuine triadic relation involves meaning, as meaning is obviously a
triadic relation. The second is that a triadic relation is inexpressible by
means of dyadic relations alone. Considerable reflexion may be required to
convince yourself of the first of these premisses, that every triadic
relation involves meaning. There will be two lines of inquiry. First, all
physical forces appear to subsist between pairs of particles. This was
assumed by Helmholtz in his original paper, On the Conservation of Forces.
Take any fact in physics of the triadic kind, by which I mean a fact which
can only be defined by simultaneous reference to three things, and you will
find there is ample evidence that it never was produced by the action of
forces on mere dyadic conditions. Thus, your right hand is that hand which
is toward the east, when you face the north with your head toward the
zenith. Three things, east, west, and up, are required to define the
difference between right and left. Consequently chemists find that those
substances which rotate the plane of polarization to the right or left can
only be produced from such [similar] active substances. They are all of such
complex constitution that they cannot have existed when the earth was very
hot, and how the first one was produced is a puzzle. It cannot have been by
the action of brute forces. For the second branch of the inquiry, you must
train yourself to the analysis of relations, beginning with such as are very
markedly triadic, gradually going on to others. In that way, you will
convince yourself thoroughly that every genuine triadic relation involves
thought or meaning. Take, for example, the relation of giving. A gives B to
C. This does not consist in A's throwing B away and its accidentally hitting
C, like the date-stone, which hit the Jinnee in the eye. If that were all,
it would not be a genuine triadic relation, but merely one dyadic relation
followed by another. There need be no motion of the thing given. Giving is a
transfer of the right of property. Now right is a matter of law, and law is
a matter of thought and meaning. I there leave the matter to your own
reflection, merely adding that, though I have inserted the word "genuine,"
yet I do not really think that necessary. I think even degenerate triadic
relations involve something like thought.

Collected Papers 1.365-367
365. Thus, the whole book being nothing but a continual exemplification of
the triad of ideas, we need linger no longer upon this preliminary
exposition of them. There is, however, one feature of them upon which it is
quite indispensable to dwell. It is that there are two distinct grades of
Secondness and three grades of Thirdness. There is a close analogy to this
in geometry. Conic sections are either the curves usually so called, or they
are pairs of straight lines. A pair of straight lines is called a degenerate
conic. So plane cubic curves are either the genuine curves of the third
order, or they are conics paired with straight lines, or they consist of
three straight lines; so that there are the two orders of degenerate cubics.
Nearly in this same way, besides genuine Secondness, there is a degenerate
sort which does not exist as such, but is only so conceived. The medieval
logicians (following a hint of Aristotle) distinguished between real
relations and relations of reason. A real relation subsists in virtue of a
fact which would be totally impossible were either of the related objects
destroyed; while a relation of reason subsists in virtue of two facts, one
only of which would disappear on the annihilation of either of the relates.
Such are all resemblances: for any two objects in nature resemble each
other, and indeed in themselves just as much as any other two; it is only
with reference to our senses and needs that one resemblance counts for more
than another. Rumford and Franklin resembled each other by virtue of being
both Americans; but either would have been just as much an American if the
other had never lived. On the other hand, the fact that Cain killed Abel
cannot be stated as a mere aggregate of two facts, one concerning Cain and
the other concerning Abel. Resemblances are not the only relations of
reason, though they have that character in an eminent degree. Contrasts and
comparisons are of the same sort. Resemblance is an identity of characters;
and this is the same as to say that the mind gathers the resembling ideas
together into one conception. Other relations of reason arise from ideas
being connected by the mind in other ways; they consist in the relation
between two parts of one complex concept, or, as we may say, in the relation
of a complex concept to itself, in respect to two of its parts. This brings
us to consider a sort of degenerate Secondness that does not fulfill the
definition of a relation of reason. Identity is the relation that everything
bears to itself: Lucullus dines with Lucullus. Again, we speak of
allurements and motives in the language of forces, as though a man suffered
compulsion from within. So with the voice of conscience: and we observe our
own feelings by a reflective sense. An echo is my own voice coming back to
answer itself. So also, we speak of the abstract quality of a thing as if it
were some second thing that the first thing possesses. But the relations of
reason and these self-relations are alike in this, that they arise from the
mind setting one part of a notion into relation to another. All degenerate
seconds may be conveniently termed internal, in contrast to external
seconds, which are constituted by external fact, and are true actions of one
thing upon another.
366. Among thirds, there are two degrees of degeneracy. The first is where
there is in the fact itself no Thirdness or mediation, but where there is
true duality; the second degree is where there is not even true Secondness
in the fact itself. Consider, first, the thirds degenerate in the first
degree. A pin fastens two things together by sticking through one and also
through the other: either might be annihilated, and the pin would continue
to stick through the one which remained. A mixture brings its ingredients
together by containing each. We may term these accidental thirds. "How did I
slay thy son?" asked the merchant, and the jinnee replied, "When thou
threwest away the date-stone, it smote my son, who was passing at the time,
on the breast, and he died forthright." Here there were two independent
facts, first that the merchant threw away the date-stone, and second that
the date-stone struck and killed the jinnee's son. Had it been aimed at him,
the case would have been different; for then there would have been a
relation of aiming which would have connected together the aimer, the thing
aimed, and the object aimed at, in one fact. What monstrous injustice and
inhumanity on the part of that jinnee to hold that poor merchant responsible
for such an accident! I remember how I wept at it, as I lay in my father's
arms and he first told me the story. It is certainly just that a man, even
though he had no evil intention, should be held responsible for the
immediate effects of his actions; but not for such as might result from them
in a sporadic case here and there, but only for such as might have been
guarded against by a reasonable rule of prudence. Nature herself often
supplies the place of the intention of a rational agent in making a
Thirdness genuine and not merely accidental; as when a spark, as third,
falling into a barrel of gunpowder, as first, causes an explosion, as
second. But how does nature do this? By virtue of an intelligible law
according to which she acts. If two forces are combined according to the
parallelogram of forces, their resultant is a real third. Yet any force may,
by the parallelogram of forces, be mathematically resolved into the sum of
two others, in an infinity of different ways. Such components, however, are
mere creations of the mind. What is the difference? As far as one isolated
event goes, there is none; the real forces are no more present in the
resultant than any components that the mathematician may imagine. But what
makes the real forces really there is the general law of nature which calls
for them, and not for any other components of the resultant. Thus,
intelligibility, or reason objectified, is what makes Thirdness genuine.
367. We now come to thirds degenerate in the second degree. The dramatist
Marlowe had something of that character of diction in which Shakespeare and
Bacon agree. This is a trivial example; but the mode of relation is
important. In natural history, intermediate types serve to bring out the
resemblance between forms whose similarity might otherwise escape attention,
or not be duly appreciated. In portraiture, photographs mediate between the
original and the likeness. In science, a diagram or analogue of the observed
fact leads on to a further analogy. The relations of reason which go to the
formation of such a triple relation need not be all resemblances. Washington
was eminently free from the faults in which most great soldiers resemble one
another. A centaur is a mixture of a man and a horse. Philadelphia lies
between New York and Washington. Such thirds may be called intermediate
thirds or thirds of comparison.

Collected Papers 1.372
372. We have seen that the mere coexistence of two singular facts
constitutes a degenerate form of dual fact; and in like manner there are two
orders of degeneracy in plural facts, for either they may consist in a mere
synthesis of facts of which the highest is dual, or they may consist in a
mere synthesis of singular facts. This explains why there should be three
classes of signs; for there is a triple connection of sign, thing signified,
cognition produced in the mind. There may be a mere relation of reason
between the sign and the thing signified; in that case the sign is an icon.
Or there may be a direct physical connection; in that case, the sign is an
index. Or there may be a relation which consists in the fact that the mind
associates the sign with its object; in that case the sign is a name [or
symbol]. Now consider the difference between a logical term, a proposition,
and an inference. A term is a mere general description, and as neither icon
nor index possesses generality, it must be a name; and it is nothing more. A
proposition is also a general description, but it differs from a term in
that it purports to be in a real relation to the fact, to be really
determined by it; thus, a proposition can only be formed of the conjunction
of a name and an index. An inference, too, contains a general description. .
. .

Collected Papers 1.383
383. Note, too, that just as we have seen that there are two orders of
Secondness, so the polar sense splits into two, and that in two ways, for
first, there is an active and a passive kind, or will and sense, and second,
there are external will and sense, in opposition to internal will
(self-control, inhibitory will) and internal sense (introspection). In like
manner, just as there are three orders of Thirdness, so there are three
kinds of synthetical consciousness. The undegenerate and really typical form
has not been made so familiar to us as the others, which have been more
completely studied by psychologists; I shall therefore mention that last.
Synthetical consciousness degenerate in the first degree, corresponding to
accidental Thirdness, is where there is an external compulsion upon us to
think things together. Association by contiguity is an instance of this; but
a still better instance is that in our first apprehension of our
experiences, we cannot choose how we will arrange our ideas in reference to
time and space, but are compelled to think certain things as nearer together
than others. It would be putting the cart before the horse to say that we
are compelled to think certain things together because they are together in
time and space; the true way of stating it is that there is an exterior
compulsion upon us to put them together in our construction of time and
space, in our perspective. Synthetical consciousness, degenerate in the
second degree, corresponding to intermediate thirds, is where we think
different feelings to be alike or different, which, since feelings in
themselves cannot be compared and therefore cannot be alike, so that to say
they are alike is merely to say that the synthetical consciousness regards
them so, comes to this, that we are internally compelled to synthesize them
or to sunder them. This kind of synthesis appears in a secondary form in
association by resemblance. But the highest kind of synthesis is what the
mind is compelled to make neither by the inward attractions of the feelings
or representations themselves, nor by a transcendental force of necessity,
but in the interest of intelligibility that is, in the interest of the
synthesizing "I think" itself; and this it does by introducing an idea not
contained in the data, which gives connections which they would not
otherwise have had. This kind of synthesis has not been sufficiently
studied, and especially the intimate relationship of its different varieties
has not been duly considered. The work of the poet or novelist is not so
utterly different from that of the scientific man. The artist introduces a
fiction; but it is not an arbitrary one; it exhibits affinities to which the
mind accords a certain approval in pronouncing them beautiful, which if it
is not exactly the same as saying that the synthesis is true, is something
of the same general kind. The geometer draws a diagram, which if not exactly
a fiction, is at least a creation, and by means of observation of that
diagram he is able to synthesize and show relations between elements which
before seemed to have no necessary connection. The realities compel us to
put some things into very close relation and others less so, in a highly
complicated, and in the [to?] sense itself unintelligible manner; but it is
the genius of the mind, that takes up all these hints of sense, adds
immensely to them, makes them precise, and shows them in intelligible form
in the intuitions of space and time. Intuition is the regarding of the
abstract in a concrete form, by the realistic hypostatization of relations;
that is the one sole method of valuable thought. Very shallow is the
prevalent notion that this is something to be avoided. You might as well say
at once that reasoning is to be avoided because it has led to so much error;
quite in the same philistine line of thought would that be; and so well in
accord with the spirit of nominalism that I wonder some one does not put it
forward. The true precept is not to abstain from hypostatization, but to do
it intelligently. . . .

Collected Papers 1.387
387. Synthetical consciousness offers a more difficult problem. Yet the
explanation of the genuine form of that consciousness, the sense of
learning, is easy enough; it is only the degenerate modes, the sense of
similarity, and the sense of real connection, which oblige us to hesitate.
With regard to these two degenerate forms, I am driven to make hypotheses.

Collected Papers 1.473
473. Every triad is either monadically degenerate, dyadically degenerate,
or genuine. A monadically degenerate triad is one which results from the
essence of three monads, its subjects. A dyadically degenerate triad is one
which results from dyads. A genuine triad is one which cannot be resolved in
any such way. That orange color is intermediate between red and yellow is a
monoidally degenerate triad. So that one given quality is a compound of two
others. So [that] red and green resemble violet more than they resemble each
other. That red is a determination of color and scarlet of red involves a
monadically degenerate triad and belongs to the class of essential triads;
yet it is properly a dyadically degenerate triad where the component dyads
are essential dyads. It is thus essential, but only indirectly essential. So
that oranges and lemons smell alike, though it is properly only a dyad, yet
may be considered as a triad, the common quality of smell being the third
subject. That a citric taste and a perfume of a cologne water kind coexist
in the lemon can only be regarded as a triad and not as a dyad. That A is
father of B and B father of C is a genuinely dyadic degenerate triad. That A
is as far north of B as B is east of C is a triad formed of two dyads of one
kind and a dyad of another kind -- (I mean the similarity of the other two,
but this is accidental). This is an almost, but not quite, genuine triad. A
is mother of B and B is wife of C. Here the two component dyads are more
independent of one another. This is a purer case of the dyadic degenerate
triad.

Collected Papers 1.481
481. So much for the first order of subdivisions of the three classes of
triads. Passing to the lower subdivisions, I find among those of the
degenerate triads nothing of particular philosophical interest; though
something may have been overlooked. But among the lower subdivisions of the
genuine triads there is an abundance.

Collected Papers 1.516-517
516. In the degenerate dyad there is a metaphysical correspondent to a
proposition; but it is a proposition whose two subjects are mere qualities.
In the first degenerate triad there is a metaphysical correspondent to a
syllogism; but it is a syllogism whose three reasons lie in mere qualities.
Thus, orange color is intermediate between red and yellow. The syllogism is
this:

Orange has in its own nature a certain indescribable but felt relation to
red;
Yellow has a similar relation to orange; as a result,
Yellow has a similar relation to red.

Now, if yellow has a relation to orange and as a result yellow has the same
relation to red, this can only be because orange has that same relation to
red.
517. In the second degenerate triad there is likewise a metaphysical
correspondent to a syllogism; but it is a syllogism whose premisses lie in
mere coexistences of dyadic facts. For example:

A is the mother of B;
B is the wife of C;
it results that A is the mother-in-law of C.

Collected Papers 1.521
521. Very wretched is the notion of [the categories] that can be conveyed
in one lecture. They must grow up in the mind, under the hot sunshine of
hard thought, daily, bright, well-focussed, and well-aimed thought; and you
must have patience, for long time is required to ripen the fruit. They are
no inventions of mine. Were they so, that would be sufficient to condemn
them. Confused notions of these elements appear in the first infancy of
philosophy, and they have never entirely been forgotten. Their fundamental
importance is noticed in the beginning of Aristotle's De Caelo, where it is
said that the Pythagoreans knew of them.

Collected Papers 1.528-529
528. Thus we have a division of seconds into those whose very being, or
Firstness, it is to be seconds, and those whose Secondness is only an
accretion. This distinction springs out of the essential elements of
Secondness. For Secondness involves Firstness. The concepts of the two kinds
of Secondness are mixed concepts composed of Secondness and Firstness. One
is the second whose very Firstness is Secondness. The other is a second
whose Secondness is second to a Firstness. The idea of mingling Firstness
and Secondness in this particular way is an idea distinct from the ideas of
Firstness and Secondness that it combines. It appears to be a conception of
an entirely different series of categories. At the same time, it is an idea
of which Firstness, Secondness, and Thirdness are component parts, since the
distinction depends on whether the two elements of Firstness and Secondness
that are united are so united as to be one or whether they remain two. This
distinction between two kinds of seconds, which is almost involved in the
very idea of a second, makes a distinction between two kinds of Secondness;
namely, the Secondness of genuine seconds, or matters, which I call genuine
Secondness, and the Secondness in which one of the seconds is only a
Firstness, which I call degenerate Secondness; so that this Secondness
really amounts to nothing but this, that a subject, in its being a second,
has a Firstness, or quality. It is to be remarked that this distinction
arose from attending to extreme cases; and consequently subdivision will be
attached to it according to the more or less essential or accidental nature
of the genuine or the degenerate Secondness. With this distinction Thirdness
has nothing to do, or at any rate has so little to do that a satisfactory
account of the distinction need not mention Thirdness.
529. I will just mention that among Firstnesses there is no distinction of
the genuine and the degenerate, while among Thirdnesses we find not only a
genuine but two distinct grades of degeneracy.

Collected Papers 1.535
535. I shall not enter into any exact analysis of these ideas. I only
wished to give you such slight glimpse as I could of the sort of questions
that busy the student of phenomenology, merely to lead up to Thirdness and
to the particular kind and aspect of Thirdness which is the sole object of
logical study. I want first to show you what genuine Thirdness is and what
are its two degenerate forms. Now we found the genuine and degenerate forms
of Secondness by considering the full ideas of first and second. Then the
genuine Secondness was found to be reaction, where first and second are both
true seconds and the Secondness is something distinct from them, while in
degenerate Secondness, or mere reference, the first is a mere first never
attaining full Secondness.

Collected Papers 1.538
538. Every sign stands for an object independent of itself; but it can only
be a sign of that object in so far as that object is itself of the nature of
a sign or thought. For the sign does not affect the object but is affected
by it; so that the object must be able to convey thought, that is, must be
of the nature of thought or of a sign. Every thought is a sign. But in the
first degree of degeneracy the Thirdness affects the object, so that this is
not of the nature of a Thirdness -- not so, at least, as far as this
operation of degenerate Thirdness is concerned. It is that the third brings
about a Secondness but does not regard that Secondness as anything more than
a fact. In short it is the operation of executing an intention. In the last
degree of degeneracy of Thirdness, there is thought, but no conveyance or
embodiment of thought at all. It is merely that a fact of which there must
be, I suppose, something like knowledge is apprehended according to a
possible idea. There is an instigation without any prompting. For example,
you look at something and say, "It is red." Well, I ask you what
justification you have for such a judgment. You reply, "I saw it was red."
Not at all. You saw nothing in the least like that. You saw an image. There
was no subject or predicate in it. It was just one unseparated image, not
resembling a proposition in the smallest particular. It instigated you to
your judgment, owing to a possibility of thought; but it never told you so.
Now in all imagination and perception there is such an operation by which
thought springs up; and its only justification is that it subsequently turns
out to be useful.

Collected Papers 2.91-92
91. In the Obsistential aspect, Originality presents itself as a Quality,
which is something which is such as it is, and is so free from Obsistence as
not even to be self-identical, or individual. Two Qualities which are alike,
as all Qualities are, are, in so far, the same Quality. Obsistence presents
itself as a Relation, which is a fact concerning a set of objects, the
Relates. A Relation is either Genuine or Degenerate. A Degenerate Relation
is a fact concerning a set of objects which consists merely in a partial
aspect of the fact that each of the Relates has its Quality. It is a
Relation of Qualities; such as that A is greater than B. Its relates may be
qualities or objects possessing qualities. It may be a Similarity, which is
a more Degenerate form, or a Difference, which is a less Degenerate form, or
it may be mixed. A Genuine Relation is one which is not necessarily involved
in its Relates having any Qualities regardless of each other. Each relate is
necessarily individual, or self-identical. Various other divisions of
relations will be made; and the nature of identity, otherness, coexistence,
and incompossibility will be specially considered.
92. Transuasion in its obsistent aspect, or Mediation, will be shown to be
subject to two degrees of degeneracy. Genuine mediation is the character of
a Sign. A Sign is anything which is related to a Second thing, its Object,
in respect to a Quality, in such a way as to bring a Third thing, its
Interpretant, into relation to the same Object, and that in such a way as to
bring a Fourth into relation to that Object in the same form, ad infinitum.
If the series is broken off, the Sign, in so far, falls short of the perfect
significant character. It is not necessary that the Interpretant should
actually exist. A being in futuro will suffice. Signs have two degrees of
Degeneracy. A Sign degenerate in the lesser degree, is an Obsistent Sign, or
Index, which is a Sign whose significance of its Object is due to its having
a genuine Relation to that Object, irrespective of the Interpretant. Such,
for example, is the exclamation "Hi!" as indicative of present danger, or a
rap at the door as indicative of a visitor. A Sign degenerate in the greater
degree is an Originalian Sign, or Icon, which is a Sign whose significant
virtue is due simply to its Quality. Such, for example, are imaginations of
how I would act under certain circumstances, as showing me how another man
would be likely to act. We say that the portrait of a person we have not
seen is convincing. So far as, on the ground merely of what I see in it, I
am led to form an idea of the person it represents, it is an Icon. But, in
fact, it is not a pure Icon, because I am greatly influenced by knowing that
it is an effect, through the artist, caused by the original's appearance,
and is thus in a genuine Obsistent relation to that original. Besides, I
know that portraits have but the slightest resemblance to their originals,
except in certain conventional respects, and after a conventional scale of
values, etc. A Genuine Sign is a Transuasional Sign, or Symbol, which is a
sign which owes its significant virtue to a character which can only be
realized by the aid of its Interpretant. Any utterance of speech is an
example. If the sounds were originally in part iconic, in part indexical,
those characters have long since lost their importance. The words only stand
for the objects they do, and signify the qualities they do, because they
will determine, in the mind of the auditor, corresponding signs. The
importance of the above divisions, although they are new, has been
acknowledged by all logicians who have seriously considered them. . . .

Collected Papers 2.265
265. In the course of the above descriptions of the classes, certain
subdivisions of some of them have been directly or indirectly referred to.
Namely, beside the normal varieties of Sinsigns, Indices, and Dicisigns,
there are others which are Replicas of Legisigns, Symbols, and Arguments,
respectively. Beside the normal varieties of Qualisigns, Icons, and Rhemes,
there are two series of others; to wit, those which are directly involved in
Sinsigns, Indices, and Dicisigns, respectively, and also those which are
indirectly involved in Legisigns, Symbols, and Arguments, respectively.
Thus, the ordinary Dicent Sinsign is exemplified by a weathercock and its
veering and by a photograph. The fact that the latter is known to be the
effect of the radiations from the object renders it an index and highly
informative. A second variety is a Replica of a Dicent Indexical Legisign.
Thus any given street cry, since its tone and theme identifies the
individual, is not a symbol, but an Indexical Legisign; and any individual
instance of it is a Replica of it which is a Dicent Sinsign. A third variety
is a Replica of a Proposition. A fourth variety is a Replica of an Argument.
Beside the normal variety of the Dicent Indexical Legisign, of which a
street cry is an example, there is a second variety, which is that sort of
proposition which has the name of a well-known individual as its predicate;
as if one is asked, "Whose statue is this?" the answer may be, "It is
Farragut." The meaning of this answer is a Dicent Indexical Legisign. A
third variety may be a premiss of an argument. A Dicent Symbol, or ordinary
proposition, in so far as it is a premiss of an Argument, takes on a new
force, and becomes a second variety of the Dicent Symbol. It would not be
worth while to go through all the varieties; but it may be well to consider
the varieties of one class more. We may take the Rhematic Indexical
Legisign. The shout of "Hullo!" is an example of the ordinary
variety--meaning, not an individual shout, but this shout "Hullo!" in
general--this type of shout. A second variety is a constituent of a Dicent
Indexical Legisign; as the word "that" in the reply, "that is Farragut." A
third variety is a particular application of a Rhematic Symbol; as the
exclamation "Hark!" A fourth and fifth variety are in the peculiar force a
general word may have in a proposition or argument. It is not impossible
that some varieties are here overlooked. It is a nice problem to say to what
class a given sign belongs; since all the circumstances of the case have to
be considered. But it is seldom requisite to be very accurate; for if one
does not locate the sign precisely, one will easily come near enough to its
character for any ordinary purpose of logic.

Collected Papers 2.274
274. A Sign, or Representamen, is a First which stands in such a genuine
triadic relation to a Second, called its Object, as to be capable of
determining a Third, called its Interpretant, to assume the same triadic
relation to its Object in which it stands itself to the same Object. The
triadic relation is genuine, that is its three members are bound together by
it in a way that does not consist in any complexus of dyadic relations. That
is the reason the Interpretant, or Third, cannot stand in a mere dyadic
relation to the Object, but must stand in such a relation to it as the
Representamen itself does. Nor can the triadic relation in which the Third
stands be merely similar to that in which the First stands, for this would
make the relation of the Third to the First a degenerate Secondness merely.
The Third must indeed stand in such a relation, and thus must be capable of
determining a Third of its own; but besides that, it must have a second
triadic relation in which the Representamen, or rather the relation thereof
to its Object, shall be its own (the Third's) Object, and must be capable of
determining a Third to this relation. All this must equally be true of the
Third's Thirds and so on endlessly; and this, and more, is involved in the
familiar idea of a Sign; and as the term Representamen is here used, nothing
more is implied. A Sign is a Representamen with a mental Interpretant.
Possibly there may be Representamens that are not Signs. Thus, if a
sunflower, in turning towards the sun, becomes by that very act fully
capable, without further condition, of reproducing a sunflower which turns
in precisely corresponding ways toward the sun, and of doing so with the
same reproductive power, the sunflower would become a Representamen of the
sun. But thought is the chief, if not the only, mode of representation.

Collected Papers 2.283
283. An Index or Seme ({se^ma}) is a Representamen whose Representative
character consists in its being an individual second. If the Secondness is
an existential relation, the Index is genuine. If the Secondness is a
reference, the Index is degenerate. A genuine Index and its Object must be
existent individuals (whether things or facts), and its immediate
Interpretant must be of the same character. But since every individual must
have characters, it follows that a genuine Index may contain a Firstness,
and so an Icon as a constituent part of it. Any individual is a degenerate
Index of its own characters.

Collected Papers 2.293-294
293. A Symbol is a law, or regularity of the indefinite future. Its
Interpretant must be of the same description; and so must be also the
complete immediate Object, or meaning. But a law necessarily governs, or "is
embodied in" individuals, and prescribes some of their qualities.
Consequently, a constituent of a Symbol may be an Index, and a constituent
may be an Icon. A man walking with a child points his arm up into the air
and says, "There is a balloon." The pointing arm is an essential part of the
symbol without which the latter would convey no information. But if the
child asks, "What is a balloon," and the man replies, "It is something like
a great big soap bubble," he makes the image a part of the symbol. Thus,
while the complete object of a symbol, that is to say, its meaning, is of
the nature of a law, it must denote an individual, and must signify a
character. A genuine symbol is a symbol that has a general meaning. There
are two kinds of degenerate symbols, the Singular Symbol whose Object is an
existent individual, and which signifies only such characters as that
individual may realize; and the Abstract Symbol, whose only Object is a
character.
294. Although the immediate Interpretant of an Index must be an Index, yet
since its Object may be the Object of an Individual [Singular] Symbol, the
Index may have such a Symbol for its indirect Interpretant. Even a genuine
Symbol may be an imperfect Interpretant of it. So an icon may have a
degenerate Index, or an Abstract Symbol, for an indirect Interpretant, and a
genuine Index or Symbol for an imperfect Interpretant.

Collected Papers 2.305
No matter of fact can be stated without the use of some sign serving as an
index. If A says to B, "There is a fire," B will ask, "Where?" Thereupon A
is forced to resort to an index, even if he only means somewhere in the real
universe, past and future. Otherwise, he has only said that there is such an
idea as fire, which would give no information, since unless it were known
already, the word "fire" would be unintelligible. If A points his finger to
the fire, his finger is dynamically connected with the fire, as much as if a
self-acting fire-alarm had directly turned it in that direction; while it
also forces the eyes of B to turn that way, his attention to be riveted upon
it, and his understanding to recognize that his question is answered. If A's
reply is, "Within a thousand yards of here," the word "here" is an index;
for it has precisely the same force as if he had pointed energetically to
the ground between him and B. Moreover, the word "yard," though it stands
for an object of a general class, is indirectly indexical, since the
yard-sticks themselves are signs of the Parliamentary Standard, and that,
not because they have similar qualities, for all the pertinent properties of
a small bar are, as far as we can perceive, the same as those of a large
one, but because each of them has been, actually or virtually, carried to
the prototype and subjected to certain dynamical operations, while the
associational compulsion calls up in our minds, when we see one of them,
various experiences, and brings us to regard them as related to something
fixed in length, though we may not have reflected that that standard is a
material bar. The above considerations might lead the reader to suppose that
indices have exclusive reference to objects of experience, and that there
would be no use for them in pure mathematics, dealing, as it does, with
ideal creations, without regard to whether they are anywhere realized or
not. But the imaginary constructions of the mathematician, and even dreams,
so far approximate to reality as to have a certain degree of fixity, in
consequence of which they can be recognized and identified as individuals.
In short, there is a degenerate form of observation which is directed to the
creations of our own minds--using the word observation in its full sense as
implying some degree of fixity and quasi-reality in the object to which it
endeavours to conform. Accordingly, we find that indices are absolutely
indispensable in mathematics; and until this truth was comprehended, all
efforts to reduce to rule the logic of triadic and higher relations failed;
while as soon as it was once grasped the problem was solved. The ordinary
letters of algebra that present no peculiarities are indices. So also are
the letters A, B, C, etc., attached to a geometrical figure. Lawyers and
others who have to state a complicated affair with precision have recourse
to letters to distinguish individuals. Letters so used are merely improved
relative pronouns. Thus, while demonstrative and personal pronouns are, as
ordinarily used, "genuine indices," relative pronouns are "degenerate
indices"; for though they may, accidentally and indirectly, refer to
existing things, they directly refer, and need only refer, to the images in
the mind which previous words have created.

Collected Papers 2.540
540. Logical induction is an induction based on examination of every
individual of the class to which the examination relates. Thus, conclusions
from a census are logical inductions. While this mode of inference is a
degenerate form of induction, it also comes into the class of dilemmatic
reasoning.

Collected Papers 3.359
359. Any character or proposition either concerns one subject, two
subjects, or a plurality of subjects. For example, one particle has mass,
two particles attract one another, a particle revolves about the line
joining two others. A fact concerning two subjects is a dual character or
relation; but a relation which is a mere combination of two independent
facts concerning the two subjects may be called degenerate, just as two
lines are called a degenerate conic. In like manner a plural character or
conjoint relation is to be called degenerate if it is a mere compound of
dual characters.

Collected Papers 3.360
360. A sign is in a conjoint relation to the thing denoted and to the mind.
If this triple relation is not of a degenerate species, the sign is related
to its object only in consequence of a mental association, and depends upon
a habit. Such signs are always abstract and general, because habits are
general rules to which the organism has become subjected. They are, for the
most part, conventional or arbitrary. They include all general words, the
main body of speech, and any mode of conveying a judgment. For the sake of
brevity I will call them tokens [= _symbols_]

Collected Papers 3.361-362 361. But if the triple relation between the
sign, its object, and the mind, is degenerate, then of the three pairs

sign object
sign mind
object mind

two at least are in dual relations which constitute the triple relation. One
of the connected pairs must consist of the sign and its object, for if the
sign were not related to its object except by the mind thinking of them
separately, it would not fulfill the function of a sign at all. Supposing,
then, the relation of the sign to its object does not lie in a mental
association, there must be a direct dual relation of the sign to its object
independent of the mind using the sign. In the second of the three cases
just spoken of, this dual relation is not degenerate, and the sign signifies
its object solely by virtue of being really connected with it. Of this
nature are all natural signs and physical symptoms. I call such a sign an
index, a pointing finger being the type of the class.
362. The third case is where the dual relation between the sign and its
object is degenerate and consists in a mere resemblance between them. I call
a sign which stands for something merely because it resembles it, an icon.
Icons are so completely substituted for their objects as hardly to be
distinguished from them. Such are the diagrams of geometry. A diagram,
indeed, so far as it has a general signification, is not a pure icon; but in
the middle part of our reasonings we forget that abstractness in great
measure, and the diagram is for us the very thing. So in contemplating a
painting, there is a moment when we lose the consciousness that it is not
the thing, the distinction of the real and the copy disappears, and it is
for the moment a pure dream -- not any particular existence, and yet not
general. At that moment we are contemplating an icon.

Collected Papers 3.392
392. The algebra of Boole affords a language by which anything may be
expressed which can be said without speaking of more than one individual at
a time. It is true that it can assert that certain characters belong to a
whole class, but only such characters as belong to each individual
separately. The logic of relatives considers statements involving two and
more individuals at once. Indices are here required. Taking, first, a
degenerate form of relation, we may write x[i]y[j] to signify that x is true
of the individual i while y is true of the individual j. If z be a relative
character z[i j] will signify that i is in that relation to j. In this way
we can express relations of considerable complexity. Thus, if

1, 2, 3,
4, 5, 6,
7, 8, 9,

are points in a plane, and l[123] signifies that 1, 2, and 3 lie on one
line, a well-known proposition of geometry may be written

Collected Papers 3.399
399. The properties of the token q must now be examined. These may all be
summed up in this, that taking any individuals i[1], i[2], i[3], etc., and
any individuals, j[1], j[2], j[3], etc., there is a collection, class, or
predicate embracing all the i's and excluding all the j's except such as are
identical with some one of the i's. This might be written

where the i's and the i' 's are the same lot of objects. This notation
presents indices of indices. The c[`]c[i[`]] shows that we are to take any
collection whatever of i's, and then any i of that collection. We are then
to do the same with the j's. We can then find a quality k such that the i
taken has it, and also such that the j taken wants it unless we can find an
i that is identical with the j taken. The necessity of some kind of notation
of this description in treating of classes collectively appears from this
consideration: that in such discourse we are neither speaking of a single
individual (as in the non-relative logic) nor of a small number of
individuals considered each for itself, but of a whole class, perhaps an
infinity of individuals. This suggests a relative term with an indefinite
series of indices as x[i j k l] . . . . Such a relative will, however, in
most, if not in all cases, be of a degenerate kind and is consequently
expressible as above. But it seems preferable to attempt a partial
decomposition of this definition. In the first place, any individual may be
considered as a class. This is written

c[i]d[k]c[j] q[k i](~q[k j] + 1[i j]).

This is the ninth icon. Next, given any class, there is another which
includes all the former excludes and excludes all the former includes. That
is,

c[l]d[k]c[i](q[l i]~q[k i] + ~q[l i]q[k i]).

This is the tenth icon. Next, given any two classes, there is a third which
includes all that either includes and excludes all that both exclude. That
is

This is the eleventh icon. Next, given any two classes, there is a class
which includes the whole of the first and any one individual of the second
which there may be not included in the first and nothing else. That is,

c[l]c[m]c[i]d[k]c[j]{q[l i]+~q[m i]+q[k i](q[k j]+~q[l j])}.

This is the twelfth icon.

Collected Papers 3.454
454. Of what use does this new logical doctrine promise to be? The first
service it may be expected to render is that of correcting a considerable
number of hasty assumptions about logic which have been allowed to affect
philosophy. In the next place, if Kant has shown that metaphysical
conceptions spring from formal logic, this great generalisation upon formal
logic must lead to a new apprehension of the metaphysical conceptions which
shall render them more adequate to the needs of science. In short, "exact"
logic will prove a stepping-stone to "exact" metaphysics. In the next place,
it must immensely widen our logical notions. For example, a class consisting
of a lot of things jumbled higgledy-piggledy must now be seen to be but a
degenerate form of the more general idea of a system. Generalisation, which
has hitherto meant passing to a larger class, must mean taking in the
conception of the whole system of which we see but a fragment, etc., etc. In
the next place, it is already evident to those who know what has already
been made out, that that speculative rhetoric, or objective logic, mentioned
at the beginning of this article, is destined to grow into a colossal
doctrine which may be expected to lead to most important philosophical
conclusions. Finally, the calculus of the new logic, which is applicable to
everything, will certainly be applied to settle certain logical questions of
extreme difficulty relating to the foundations of mathematics. Whether or
not it can lead to any method of discovering methods in mathematics it is
difficult to say. Such a thing is conceivable.

Collected Papers 3.554
554. I wish I knew with certainty the precise origin of the definition of
mathematics as the science of quantity. It certainly cannot be Greek,
because the Greeks were advanced in projective geometry, whose problems are
such as these: whether or not four points obtained in a given way lie in one
plane; whether or not four planes have a point in common; whether or not two
rays (or unlimited straight lines) intersect, and the like -- problems which
have nothing to do with quantity, as such. Aristotle names, as the subjects
of mathematical study, quantity and continuity. But though he never gives a
formal definition of mathematics, he makes quite clear, in more than a dozen
places, his view that mathematics ought not to be defined by the things
which it studies but by its peculiar mode and degree of abstractness.
Precisely what he conceives this to be it would require me to go too far
into the technicalities of his philosophy to explain; and I do not suppose
anybody would today regard the details of his opinion as important for my
purpose. Geometry, arithmetic, astronomy, and music were, in the Roman
schools of the fifth century and earlier, recognized as the four branches of
mathematics. And we find Bo thius (A.D. 500) defining them as the arts which
relate, not to quantity, but to quantities, or quanta. What this would seem
to imply is, that mathematics is the foundation of the minutely exact
sciences; but really it is not worth our while, for the present purpose, to
ascertain what the schoolmasters of that degenerate age conceived
mathematics to be.

Collected Papers 4.147
Radiating from each point of the plane is a continuity of lines. Each of
these has upon it its two absolute points (possibly imaginary, and even
possibly coincident); and assuming these to be continuous, they form a curve
which, being cut in two points only by any one line, is of the second order.
That is, it is a conic section, though it may be an imaginary or even
degenerate one.
Now as the foot has different lengths in different countries, so the
ratios of units of lengths along different lines in the plane is somewhat
arbitrary. But the measurement is so made that first, every point infinitely
distant from another along a straight line is also infinitely distant along
any broken line; and second, if two straight lines intersect at a point, A,
on the absolute conic and respectively cut it again at B and C; and from D,
any point collinear with B and C, two straight lines be drawn, the segments
of the first two lines, EF and GH, which these cut off, are equal. I omit
the geometrical proof that this involves no inconsistency. This proposition
enables us to compare any two lengths.

Collected Papers 4.256
256. If the relation between a function and its arguments is such that one
of the latter may take any value for every set of the values of the others
without altering the function, the function may be said to be invariable
with that argument. If the function can take any value, whatever values be
assigned to the arguments, it may be said to be independent of the
arguments. In either of these cases, the function may be called a degenerate
function.

Collected Papers 5.66
66. Category the First is the Idea of that which is such as it is
regardless of anything else. That is to say, it is a Quality of Feeling.

Collected Papers 5.68-69
68. Category the First owing to its Extremely Rudimentary character is not
susceptible of any degenerate or weakened modification.
69. Category the Second has a Degenerate Form, in which there is Secondness
indeed, but a weak or Secondary Secondness that is not in the pair in its
own quality, but belongs to it only in a certain respect. Moreover, this
degeneracy need not be absolute but may be only approximative. Thus a genus
characterized by Reaction will by the determination of its essential
character split into two species, one a species where the secondness is
strong, the other a species where the secondness is weak, and the strong
species will subdivide into two that will be similarly related, without any
corresponding subdivision of the weak species. For example, Psychological
Reaction splits into Willing, where the Secondness is strong, and Sensation,
where it is weak; and Willing again subdivides into Active Willing and
Inhibitive Willing, to which last dichotomy nothing in Sensation
corresponds. But it must be confessed that subdivision, as such, involves
something more than the second category.

Collected Papers 5.71
71. The most degenerate Thirdness is where we conceive a mere Quality of
Feeling, or Firstness, to represent itself to itself as Representation.
Such, for example, would be Pure Self-Consciousness, which might be roughly
described as a mere feeling that has a dark instinct of being a germ of
thought. This sounds nonsensical, I grant. Yet something can be done toward
rendering it comprehensible.
72. The relatively degenerate forms of the Third category do not fall into
a catena, like those of the Second. What we find is this. Taking any class
in whose essential idea the predominant element is Thirdness, or
Representation, the self-development of that essential idea -- which
development, let me say, is not to be compassed by any amount of mere "hard
thinking," but only by an elaborate process founded upon experience and
reason combined -- results in a trichotomy giving rise to three sub-classes,
or genera, involving respectively a relatively genuine thirdness, a
relatively reactional thirdness or thirdness of the lesser degree of
degeneracy, and a relatively qualitative thirdness or thirdness of the last
degeneracy. This last may subdivide, and its species may even be governed by
the three categories, but it will not subdivide, in the manner which we are
considering, by the essential determinations of its conception. The genus
corresponding to the lesser degree of degeneracy, the reactionally
degenerate genus, will subdivide after the manner of the Second category,
forming a catena; while the genus of relatively genuine Thirdness will
subdivide by Trichotomy just like that from which it resulted. Only as the
division proceeds, the subdivisions become harder and harder to discern.

Collected Papers 5.73
Of these three genera of representamens, the Icon is the Qualitatively
degenerate, the Index the Reactionally degenerate, while the Symbol is the
relatively genuine genus.

Collected Papers 5.75-76
75. It is quite otherwise with the Index. Here is a reactional sign, which
is such by virtue of a real connection with its object. Then the question
arises is this dual character in the Index, so that it has two elements, by
virtue of the one serving as a substitute for the particular object it does,
while the other is an involved icon that represents the representamen itself
regarded as a quality of the object -- or is there really no such dual
character in the index, so that it merely denotes whatever object it happens
to be really connected with just as the icon represents whatever object it
happens really to resemble? Of the former, the relatively genuine form of
Index, the hygrometer, is an example. Its connection with the weather is
dualistic, so that by an involved icon, it actually conveys information. On
the other hand any mere land-mark by which a particular thing may be
recognized because it is as a matter of fact associated with that thing, a
proper name without signification, a pointing finger, is a degenerate index.
Horatio Greenough, who designed Bunker Hill Monument, tells us in his book
that he meant it to say simply "Here!" It just stands on that ground and
plainly is not movable. So if we are looking for the battle-field, it will
tell us whither to direct our steps.
76. The Symbol, or relatively genuine form of Representamen, divides by
Trichotomy into the Term, the Proposition, and the Argument. The Term
corresponds to the Icon and to the degenerate Index. It does excite an icon
in the imagination. The proposition conveys definite information like the
genuine index, by having two parts of which the function of the one is to
indicate the object meant, while that of the other is to represent the
representamen by exciting an icon of its quality. The argument is a
representamen which does not leave the interpretant to be determined as it
may by the person to whom the symbol is addressed, but separately represents
what is the interpreting representation that it is intended to determine.
This interpreting representation is, of course, the conclusion. It would be
interesting to push these illustrations further; but I can linger nowhere.
As soon as a subject begins to be interesting I am obliged to pass on to
another.

Collected Papers 5.89
89. The best way of satisfying oneself whether Thirdness is elementary or
not -- at least, it would be the best way for me, who had in the first place
a natural aptitude for logical analysis which has been in constant training
all my life long (and I rather think it would be the best way for anybody
provided he ruminates over his analysis, returns to it again and again, and
criticizes it severely and sincerely, until he reaches a complete insight
into the analysis) -- the best way, I say, is to take the idea of
representation, say the idea of the fact that the object, A, is represented
in the representation, B, so as to determine the interpretation, C: to take
this idea and endeavor to state what it consists in without introducing the
idea of Thirdness at all if possible, or, if you find that impossible, to
see what is the minimum or most degenerate form of Thirdness which will
answer the purpose.
Then, having exercised yourself on that problem, take another idea in
which, according to my views, Thirdness takes a more degenerate form. Try
your hand at a logical analysis of the Fact that A gives B to C.
Then pass to a case in which Thirdness takes a still more degenerate form,
as for example the idea of "A and B." What is at once A and B involves the
idea of three variables. Putting it mathematically, it is Z = XY, which is
the equation of the simpler of the two hyperboloids, the two-sheeted one, as
it is called.

Collected Papers 5.103
103. In another respect, however, the definition represents a very
degenerate sort of generality. None of the scholastic logics fails to
explain that sol is a general term; because although there happens to be but
one sun yet the term sol aptum natum est dici de multis. But that is most
inadequately expressed. If sol is apt to be predicated of many, it is apt to
be predicated of any multitude however great, and since there is no maximum
multitude, those objects, of which it is fit to be predicated, form an
aggregate that exceeds all multitude. Take any two possible objects that
might be called suns and, however much alike they may be, any multitude
whatsoever of intermediate suns are alternatively possible, and therefore as
before these intermediate possible suns transcend all multitude. In short,
the idea of a general involves the idea of possible variations which no
multitude of existent things could exhaust but would leave between any two
not merely many possibilities, but possibilities absolutely beyond all
multitude.

Collected Papers 5.542
In order to handle this question, it is necessary to draw a distinction.
Every belief is belief in a proposition. Now every proposition has its
predicate which expresses what is believed, and its subjects which express
of what it is believed. The grammarians of today prefer to say that a
sentence has but one subject, which is put in the nominative. But from a
logical point of view the terminology of the older grammarians was better,
who spoke of the subject nominative and the subject accusative. I do not
know that they spoke of the subject dative; but in the proposition, "Anthony
gave a ring to Cleopatra," Cleopatra is as much a subject of what is meant
and expressed as is the ring or Anthony. A proposition, then, has one
predicate and any number of subjects. The subjects are either names of
objects well known to the utterer and to the interpreter of the proposition
(otherwise he could not interpret it) or they are virtually almost
directions how to proceed to gain acquaintance with what is referred to.
Thus, in the sentence "Every man dies," "Every man" implies that the
interpreter is at liberty to pick out a man and consider the proposition as
applying to him. In the proposition "Anthony gave a ring to Cleopatra," if
the interpreter asks, What ring? the answer is that the indefinite article
shows that it is a ring which might have been pointed out to the interpreter
if he had been on the spot; and that the proposition is only asserted of the
suitably chosen ring. The predicate on the other hand is a word or phrase
which will call up in the memory or imagination of the interpreter images of
things such as he has seen or imagined and may see again. Thus, "gave" is
the predicate of the last proposition; and it conveys its meaning because
the interpreter has had many experiences in which gifts were made; and a
sort of composite photograph of them appears in his imagination. I am told
that "Saccharin is 500 times as sweet as cane-sugar." But I never heard of
saccharin. On inquiry, I find it is the sulphimide of orthosulphobenzoic
acid; that is, it is phthalimide in which one CO group is replaced by SO[2].
I can see on paper that there might be such a body. That it is "500 times
sweeter than sugar" produces a rather confused idea of a very familiar
general kind. What I am to expect is expressed by the predicate, while the
subjects inform me on what occasion I am to expect it. Diogenes Laertius,
Suidas, Plutarch, and an anonymous biographer tell us that Aristotle was
unable to pronounce the letter R. I place Aristotle perfectly, of course. He
is the author of works I often read and profoundly admire and whose fame far
surpasses that of any other logician -- The Prince of Philosophers. I have
also met people who could not pronounce R; but in other respects they did
not seem to be much like Aristotle -- not even Dundreary. Should I meet him
in the Elysian Fields, I shall know what to expect. That is an impossible
supposition; but should I ever meet a great logician, spindle-shanked and
pig-eyed, who cannot pronounce R, I shall be interested to see whether he
has other characteristics of Aristotle. This example has been selected as
one which should seem to a superficial eye to involve no gleam of
expectation; and if this testimony of four respectable witnesses, as
independent as under the circumstances they could be, is destined never to
receive confirmation nor contradiction, nor in any other way to have its
probable consequences confronted by future experience, then in truth no
expectation does it carry. In that case, it is an idle tale that might, for
any practical purpose, have been as well the creation of some ironical poet.
In that case, it is, properly speaking, no contribution to knowledge, for at
least it is only probability, and probability cannot be reckoned as
knowledge, unless it is destined to be indefinitely heightened in the
future. Knowledge which should have no possible bearing upon any future
experience -- bring no expectation whatever -- would be information
concerning a dream. But in truth no such thing can be presumed of any
knowledge. We expect that in time it will produce, or reinforce, or weaken
some definite expectation. Give science only a hundred more centuries of
increase in geometrical progression, and she may be expected to find that
the sound waves of Aristotle's voice have somehow recorded themselves. If
not, it were better to hand the reports over to the poets to make something
pretty of, and thus turn them to some human use. But the right thing to do
is to expect the verification. It is the degenerate pronunciation that is to
be expected; the occasion is when Aristotle's voice shall become virtually
heard again or when we shall have some other information which shall confirm
or refute those reports.

Collected Papers 5.71 Footnote P1 p 49
^P1 This gives an idea of the second degree of degenerate Thirdness. Those
of you who have read Professor Royce's Supplementary Essay [in The World and
the Individual, vol. 1, p. 505, n. 1] will have remarked that he avoids this
result, which does not suit his philosophy, by not allowing his map to be
continuous. But to exclude continuity is to exclude what is best and most
living in Hegel -- from the alternative "a" version.

Collected Papers 6.303-304
303. All three modes of evolution are composed of the same general
elements. Agapasm exhibits them the most clearly. The good result is here
brought to pass, first, by the bestowal of spontaneous energy by the parent
upon the offspring, and, second, by the disposition of the latter to catch
the general idea of those about it and thus to subserve the general purpose.
In order to express the relation that tychasm and anancasm bear to agapasm
let me borrow a word from geometry. An ellipse crossed by a straight line is
a sort of cubic curve; for a cubic is a curve which is cut thrice by a
straight line; now a straight line might cut the ellipse twice and its
associated straight line a third time. Still the ellipse with the straight
line across it would not have the characteristics of a cubic. It would have,
for instance, no contrary flexure, which no true cubic wants; and it would
have two nodes, which no true cubic has. The geometers say that it is a
degenerate cubic. Just so, tychasm and anancasm are degenerate forms of
agapasm.
304. Men who seek to reconcile the Darwinian idea with Christianity will
remark that tychastic evolution, like the agapastic, depends upon a
reproductive creation, the forms preserved being those that use the
spontaneity conferred upon them in such wise as to be drawn into harmony
with their original, quite after the Christian scheme. Very good! This only
shows that just as love cannot have a contrary, but must embrace what is
most opposed to it, as a degenerate case of it, so tychasm is a kind of
agapasm. Only, in the tychastic evolution, progress is solely owing to the
distribution of the napkin-hidden talent of the rejected servant among those
not rejected, just as ruined gamesters leave their money on the table to
make those not yet ruined so much the richer. It makes the felicity of the
lambs just the damnation of the goats, transposed to the other side of the
equation. In genuine agapasm, on the other hand, advance takes place by
virtue of a positive sympathy among the created springing from continuity of
mind. This is the idea which tychasticism knows not how to manage.

Collected Papers 6.322
322. For forty years, that is, since the beginning of the year 1867, I have
been constantly on the alert to find a genuine triadic relation -- that is,
one that does not consist in a mere collocation of dyadic relations, or the
negative of such, etc. (I prefer not to attempt a perfectly definite
definition) -- which is not either an intellectual relation or a relation
concerned with the less comprehensible phenomena of life. I have not met
with one which could not reasonably be supposed to belong to one or other of
these two classes. As a case as nearly brute and inorganic as any, I may
mention the form of relationship involved in any screw-form which is
definitely of the right-hand, or occidental, mode, or is definitely of the
Japanese, or left-handed, mode. Such a relation exists in every carbon-atom
whose four valencies are saturated by combination with four atoms of as many
different kinds. But where the action of chance determines whether the screw
be a right-handed or a left-handed one, the two forms will, in the long run,
be produced in equal proportions, and the general result will not be
definitely, or decisively, of either kind. We know no case of a definitely
right-handed or left-handed screw-phenomenon, where the decision is not
certainly due to the intervention of a definitely one-sided screw in the
conditions of that decision, except in cases where the choice of a living
being determines it; as when Pasteur picked out under the microscope the two
kinds of crystals of a tartrate, and shoved those of one kind to the right
and those of the other kind to the left. We do not know the mechanism of
such choice, and cannot say whether it be determined by an antecedent
separation of left-handed screws from right-handed screws or not. No doubt,
all that chance is competent to destroy, it may, once in a long, long time,
produce; but it is a question whether absolute chance -- pure tychism --
ought not to be regarded as a product of freedom, and therefore of life, not
necessarily physiological. It could not be caused, apparently, by the
inorganic action of dynamical law. For the only way in which the laws of
dynamics involve triadic relations is by their reference to second
differentials of positions. But though a second differential generally
involves a triadic relation, yet owing to the law of the conservation of
energy, which has been sufficiently proved for purely inorganic phenomena,
the dynamic laws for such phenomena are expressible in terms of first
differentials. It is, therefore, a non-genuine, or, as I phrase it, a
"degenerate" form of triadic relationship which is involved in such case. In
short, the problem of how genuine triadic relationships first arose in the
world is a better, because more definite, formulation of the problem of how
life first came about; and no explanation has ever been offered except that
of pure chance, which we must suspect to be no explanation, owing to the
suspicion that pure chance may itself be a vital phenomenon. In that case,
life in the physiological sense would be due to life in the metaphysical
sense. Of course, the fact that a given individual has been persuaded of the
truth of a proposition is the very slenderest possible argument for its
truth; nevertheless, the fact that I, a person of the strongest possible
physicistic prejudices, should, as the result of forty years of
questionings, have been brought to the deep conviction that there is some
essentially and irreducibly other element in the universe than pure dynamism
may have sufficient interest to excuse my devoting a single sentence to its
expression. For you may be sure that I had reasons that withstood severe,
not to say hostile criticism; and if I live to do it, I shall embody them in
a volume.

Collected Papers 8.330-332
330. The type of an idea of Secondness is the experience of effort,
prescinded from the idea of a purpose. It may be said that there is no such
experience, that a purpose is always in view as long as the effort is
cognized. This may be open to doubt; for in sustained effort we soon let the
purpose drop out of view. However, I abstain from psychology which has
nothing to do with ideoscopy [i.e. phenomenology]. The existence of the word
effort is sufficient proof that people think they have such an idea; and
that is enough. The experience of effort cannot exist without the experience
of resistance. Effort only is effort by virtue of its being opposed; and no
third element enters. Note that I speak of the experience, not of the
feeling, of effort. Imagine yourself to be seated alone at night in the
basket of a balloon, far above earth, calmly enjoying the absolute calm and
stillness. Suddenly the piercing shriek of a steam-whistle breaks upon you,
and continues for a good while. The impression of stillness was an idea of
Firstness, a quality of feeling. The piercing whistle does not allow you to
think or do anything but suffer. So that too is absolutely simple. Another
Firstness. But the breaking of the silence by the noise was an experience.
The person in his inertness identifies himself with the precedent state of
feeling, and the new feeling which comes in spite of him is the non-ego. He
has a two-sided consciousness of an ego and a non-ego. That consciousness of
the action of a new feeling in destroying the old feeling is what I call an
experience. Experience generally is what the course of life has compelled me
to think. Secondness is either genuine or degenerate. There are many degrees
of genuineness. Generally speaking genuine secondness consists in one thing
acting upon another, -- brute action. I say brute, because so far as the
idea of any law or reason comes in, Thirdness comes in. When a stone falls
to the ground, the law of gravitation does not act to make it fall. The law
of gravitation is the judge upon the bench who may pronounce the law till
doomsday, but unless the strong arm of the law, the brutal sheriff, gives
effect to the law, it amounts to nothing. True, the judge can create a
sheriff if need be; but he must have one. The stone's actually falling is
purely the affair of the stone and the earth at the time. This is a case of
reaction. So is existence which is the mode of being of that which reacts
with other things. But there is also action without reaction. Such is the
action of the previous upon the subsequent. It is a difficult question
whether the idea of this one-sided determination is a pure idea of
secondness or whether it involves thirdness. At present, the former view
seems to me correct. I suppose that when Kant made Time a form of the
internal sense alone, he was influenced by some such considerations as the
following. The relation between the previous and the subsequent consists in
the previous being determinate and fixed for the subsequent, and the
subsequent being indeterminate for the previous. But indeterminacy belongs
only to ideas; the existent is determinate in every respect; and this is
just what the law of causation consists in. Accordingly, the relation of
time concerns only ideas. It may also be argued that, according to the law
of the conservation of energy, there is nothing in the physical universe
corresponding to our idea that the previous determines the subsequent in any
way in which the subsequent does not determine the previous. For, according
to that law, all that happens in the physical universe consists in the
exchange of just so much vis viva 1/2m(ds/dt).2/ for so much displacement.
Now the square of a negative quantity being positive, it follows that if all
the velocities were reversed at any instant, everything would go on just the
same, only time going backward as it were. Everything that had happened
would happen again in reversed order. These seem to me to be strong
arguments to prove that temporal causation (a very different thing from
physical dynamic action) is an action upon ideas and not upon existents. But
since our idea of the past is precisely the idea of that which is absolutely
determinate, fixed, fait accompli, and dead, as against the future which is
living, plastic, and determinable, it appears to me that the idea of
one-sided action, in so far as it concerns the being of the determinate, is
a pure idea of Secondness; and I think that great errors of metaphysics are
due to looking at the future as something that will have been past. I cannot
admit that the idea of the future can be so translated into the Secundal
ideas of the past. To say that a given kind of event never will happen is to
deny that there is any date at which its happening will be past; but it is
not equivalent to any affirmation about a past relative to any assignable
date. When we pass from the idea of an event to saying that it never will
happen, or will happen in endless repetition, or introduce in any way the
idea of endless repetition, I will say the idea is mellonized ({mell_n}},
about to be, do, or suffer). When I conceive a fact as acting but not
capable of being acted upon, I will say that it is parelelythose
({parellyth_s}, past) and the mode of being which consists in such action I
will call parelelythosine (-ine = {einai}, being); I regard the former as an
idea of Thirdness, the latter as an idea of Secondness. I consider the idea
of any dyadic relation not involving any third as an idea of Secondness; and
I should not call any completely degenerate except the relation of identity.
But similarity which is the only possible identity of Firsts is very near to
that. Dyadic relations have been classified by me in a great variety of
ways; but the most important are, first, with regard to the nature of the
Second in itself and, second, with regard to the nature of its First. The
Second, or Relate,is, in itself, either a Referate, if it is intrinsically a
possibility, such as a quality, or it is a Revelate if it is of its own
nature an Existent. In respect to its First, the Second is divisible either
in regard to the dynamic first or to the immediate first. In regard to its
dynamic first, a Second is determined either by virtue of its own intrinsic
nature, or by virtue of a real relation to that second (an action). Its
immediate second is either a Quality or an Existent.
331. I now come to Thirdness. To me, who have for forty years considered
the matter from every point of view that I could discover, the inadequacy of
Secondness to cover all that is in our minds is so evident that I scarce
know how to begin to persuade any person of it who is not already convinced
of it. Yet I see a great many thinkers who are trying to construct a system
without putting any thirdness into it. Among them are some of my best
friends who acknowledge themselves indebted to me for ideas but have never
learned the principal lesson. Very well. It is highly proper that Secondness
should be searched to its very bottom. Thus only can the indispensableness
and irreducibility of thirdness be made out, although for him who has the
mind to grasp it, it is sufficient to say that no branching of a line can
result from putting one line on the end of another. My friend Schroeder fell
in love with my algebra of dyadic relations. The few pages I gave to it in
my Note B in the 'Studies in Logic by Members of the Johns Hopkins
University' were proportionate to its importance. His book is profound, but
its profundity only makes it more clear that Secondness cannot compass
Thirdness. (He is careful to avoid ever saying that it can, but he does go
so far as to say that Secondness is the more important. So it is,
considering that Thirdness cannot be understood without Secondness. But as
to its application, it is so inferior to Thirdness as to be in that aspect
quite in a different world.) Even in the most degenerate form of Thirdness,
and thirdness has two grades of degeneracy, something may be detected which
is not mere secondness. If you take any ordinary triadic relation, you will
always find a mental element in it. Brute action is secondness, any
mentality involves thirdness. Analyze for instance the relation involved in
'A gives B to C.' Now what is giving? It does not consist [in] A's putting B
away from him and C's subsequently taking B up. It is not necessary that any
material transfer should take place. It consists in A's making C the
possessor according to Law. There must be some kind of law before there can
be any kind of giving, -- be it but the law of the strongest. But now
suppose that giving did consist merely in A's laying down the B which C
subsequently picks up. That would be a degenerate form of Thirdness in which
the thirdness is externally appended. In A's putting away B, there is no
thirdness. In C's taking B, there is no thirdness. But if you say that these
two acts constitute a single operation by virtue of the identity of the B,
you transcend the mere brute fact, you introduce a mental element . . . .
The criticism which I make on [my] algebra of dyadic relations, with which I
am by no means in love, though I think it is a pretty thing, is that the
very triadic relations which it does not recognize, it does itself employ.
For every combination of relatives to make a new relative is a triadic
relation irreducible to dyadic relations. Its inadequacy is shown in other
ways, but in this way it is in a conflict with itself if it be regarded, as
I never did regard it, as sufficient for the expression of all relations. My
universal algebra of relations, with the subjacent indices and d and c, is
susceptible of being enlarged so as to comprise everything; and so, still
better, though not to ideal perfection, is the system of existential graphs.
332. I have not sufficiently applied myself to the study of the degenerate
forms of Thirdness, though I think I see that it has two distinct grades of
degeneracy. In its genuine form, Thirdness is the triadic relation existing
between a sign, its object, and the interpreting thought, itself a sign,
considered as constituting the mode of being of a sign. A sign mediates
between the interpretant sign and its object. Taking sign in its broadest
sense, its interpretant is not necessarily a sign. Any concept is a sign, of
course. Ockham, Hobbes, and Leibniz have sufficiently said that. But we may
take a sign in so broad a sense that the interpretant of it is not a
thought, but an action or experience, or we may even so enlarge the meaning
of sign that its interpretant is a mere quality of feeling. A Third is
something which brings a First into relation to a Second. A sign is a sort
of Third. How shall we characterize it? Shall we say that a Sign brings a
Second, its Object, into cognitive relation to a Third? That a Sign brings a
Second into the same relation to a first in which it stands itself to that
First? If we insist on consciousness, we must say what we mean by
consciousness of an object. Shall we say we mean Feeling? Shall we say we
mean association, or Habit? These are, on the face of them, psychological
distinctions, which I am particular to avoid. What is the essential
difference between a sign that is communicated to a mind, and one that is
not so communicated? If the question were simply what we do mean by a sign,
it might soon be resolved. But that is not the point. We are in the
situation of a zo_logist who wants to know what ought to be the meaning of
"fish" in order to make fishes one of the great classes of vertebrates. It
appears to me that the essential function of a sign is to render inefficient
relations efficient, -- not to set them into action, but to establish a
habit or general rule whereby they will act on occasion. According to the
physical doctrine, nothing ever happens but the continued rectilinear
velocities with the accelerations that accompany different relative
positions of the particles. All other relations, of which we know so many,
are inefficient. Knowledge in some way renders them efficient; and a sign is
something by knowing which we know something more. With the exception of
knowledge, in the present instant, of the contents of consciousness in that
instant (the existence of which knowledge is open to doubt) all our thought
and knowledge is by signs. A sign therefore is an object which is in
relation to its object on the one hand and to an interpretant on the other,
in such a way as to bring the interpretant into a relation to the object,
corresponding to its own relation to the object. I might say 'similar to its
own' for a correspondence consists in a similarity; but perhaps
correspondence is narrower.

In as much as relations are nothing but aggregates, sets, or logical sums
of elementary relations (or ordered tuples), we may do with relations the
whole variety of familiar set-theoretic operations that we are accustomed
to do carry out with the class of sets from which relations inherit their
properties as sets, for instance: complementation, intersection, union,
relative difference, and all the rest.

Here are a few more bits of terminology that come up at this point:

The "cardinality" |A| of a set A is, roughly speaking,
nothing more or less than the number of elements in A.
In the sorts of finite cases that presently occupy us,
roughly speaking will be ready enough.

The "power set" Pow(A) of a set A is the set of all subsets of A.
Hence, in so far as it concerns the finite case, |Pow(A)| = 2^|A|.

Let's use "bang-bang brackets", excuse my Anglish, taking the form "!...!",
as a font-shifting device, to transcribe Fraktur, Greek, or script letters,
for instance, !L! for script L, !P! for Greek Pi, !S! for Greek Sigma, etc.
If we start running out of letters, I will shift to using "scrip brackets",
taking the form "$...$", for script letters, but I'd really prefer not to.

----quote----------
"Genuine" is a term of art in Peirce's philosophy, and is not equivalent to
"real" as opposed to "apparent". I am distributing separately a collection of
passages from the CP compiled some time ago. Bear in mind that they were
acquired by a string search, and I have only minimally edited them for
legibility. The first paragraph is the same as the one Gary quoted, but
contains a little more, I believe..
----end quote-----

Thanks for the collection of passages, Joe. I want to encourage readers of
the list to look through them. My expectation is that "genuine" will be found
to have various meanings or uses in Peirce, as elsewhere in philosophy and
ordinary speech. Still, even if "genuine" in the first passage, the one I
commented on, should require some special interpretation not evident from the
limited context, I think it doubtful that Peirce can avoid something very
similar to the distinction I attributed in relation to the passage.

We need only consider identity itself. In logic identity is generally
considered to be a binary or dyadic relation. It is written as a two-placed
prdicate, certainly, and so employed. But Peirce recommends the concept of
teridentity, in which the identity predicate is not so limited. So, to put
the matter directly, we have been asking, in part, in this thread whether
identity is or is properly considered a dyadic or binary relation or instead
in some other way akin to Peirce's conception of teridentity. Opinions differ
on the question. So, there is at least the possibility here that we will find
out that some conceptions of identity are defective, or limited, specifically
with regard to their valency. (Say, that we do better not to regard identity
as a dyadic or binary relation.) That is to suggest the possibility that what
seems to be a dyadic (or triadic) relation has been misconstrued or
misunderstood and is better understood in the alternative way. Whether this
is thought of as a matter of "genuine" (meaning "really" triadic of dyadic
relations) or in some other way is less important than the possibility that
we may be able to decide, only after some considerable inquiry, to construe
identity in one way or the other. If this is a possible result and course of
inquiry in the case of identity, then we have little reason to think that it
would not also turn out similarly with other relations.

Let's see what we come up with from the passages you supply. Thanks for your
interest in this and related threads.

I would like to echo Howard's thanks for your supplying therelevant
quotations, Joe; and to thank you Howard for your
thoughtful response to my post which, along with your most recent one to
Joe--and the quotations already mentioned--I'll
be digesting most all of the free hours I have today (which,
unfortunately, aren't many).

Let me say that both your comments have already suggested to me that
this is a far more subtle matter than I had once
imagined. I think you are quite right about that, Joe. Also, I am
beginning to follow more fully some parts of the identity-teridentity
problematic which I had not previously, thanks to your recent comments,
Howard. (More on this after the rich meal of quotes and
comments has been digested.)

Let me quickly correct an error in my first post. I suggested that
chirality might be seen as a dyadic relationship, when
it seems to me now that it is no relationship at all, but a quality, a
firstness, and that handedness ought be analyzed just
as Peirce did in CP 1.345. (BYW, that elementary particles skew to the
right or to the left is another of those discoveries of
twentienth century science that Peirce anticipated--I'm not suggesting
that this is the passage in which he did so.)

I am supplementing Joe's quotations with one that I consider most
relevant to this discussion. It is a continuation
of the passage which both Joe and I have already quoted (actually Joe's,
which completed the paragraph, while
mine did not). CP 1346 offers "[t}he other premiss of the argument that
geniune triadic relations can never be built
of dyadic relations."

Gary

Peirce: CP 1.346 Cross-Ref:++
346. The other premiss of the argument that genuine triadic
relations can never be built of dyadic relations and of qualities is
easily shown. In existential graphs, a spot with one tail -- X
represents a quality, a spot with two tails -- R -- a dyadic relation.+1
Joining the ends of two tails is also a dyadic relation. But you can
never by such joining make a graph with three tails. You may think that
a node connecting three lines of identity Y is not a triadic idea. But
analysis will show that it is so. I see a man on Monday. On Tuesday I
see a man, and I exclaim, "Why, that is the very man I saw on Monday."
We may say, with sufficient accuracy, that I directly experienced the
identity. On Wednesday I see a man and I say, "That is the same man I
saw on Tuesday, and consequently is the same I saw on Monday." There is
a recognition of triadic identity; but it is only brought about as a
conclusion from two premisses, which is itself a triadic relation. If I
see two men at once, I cannot by any such direct experience identify
both of them with a man I saw before. I can only identify them if I
regard them, not as the very same, but as two different manifestations
of the same man. But the idea of manifestation is the idea of a sign.
Now a sign is something, A, which denotes some fact or object, B, to
some interpretant thought, C.

>Peirce-l,
>
>Here follows some comments and analysis of the Peirce quote helpfully
>supplied by Gary Richmond. Generally I aim to keep an eye on the thesis of
>the non-reducibility of triadic relations.

Fixing a typo -- or is it a tokum? --
I think that we can begin to
advance from this point:

In as much as relations are nothing but aggregates, sets, or logical sums
of elementary relations (or ordered tuples), we may do with relations the
whole variety of familiar set-theoretic operations that we are accustomed
to carry out with the category of sets from which relations inherit their
properties as sets, for instance: complementation, intersection, union,
asymmetric difference, symmetric difference, and all the rest.

As a general rule, especially in the earliest papers, Peirce will analogi=
ze
intersections with products, unions with sums, whose sigils are !P! and !=
S!,
respectively, and he also dubs them the "non-relative aggregate or sum" a=
nd
the "non-relative composite or product", respectively, as best I can reca=
ll.

By way of acquiring some practical experience with the materials and tool=
s in
this shop, let us devise a concrete example, whose study should be suffic=
ient.

For the immediate if not exactly the unmitigated future,
we will contemplate only those sorts of operations that
are defined on relations of the same arities or types.
It is possible to get fancier about this, speaking of
formal objects called "relation complexes", but then
we would have to be very careful about what we mean
by expressions like "i + i:j + i:j:k", and how the
sets !L!_1, !L!_2, !L!_3, ... "embed" or "inject"
themselves into a more encompassing family !L!.
I judge that this'd be too distracting at this
stage of the game, so let's not go there, yet.

I am writing this from memory in deep cold-storage --
modulo memory and fallibility I think this is more
or less how it goes, but I may need to go back and
retrieve Peirce's actual notations at some point.

We have now seen enough of the ordinary set-theoretic,
the "non-relative" operations on relations to get the
general idea, but we haven't yet touched the relative
operations on relations. This is apparently so terra
incognita even for many who speak so interpidly about
the compositions, the decompositions, the productions,
and the reductions of all kinds of relations that I'm
sure that it would come as a shock to them when first
they step off their enfolding maps onto terra firma.

So let us amble onward with freshly opened eyes, as if
seeing our place under the sun for the very first time.

The operation on 2-adic relations that Peirce knew under the
names of "relative multiplication" or "relative product" can
be defined, I say, 'DEFINED' in the following way, where for
the sake of a beginnning account I will give this first time
an informal but perfectly adequate version of the definition.

To compute UV, in general, where U and V are 2-adic relations,
simply multiply out the two sums in the ordinary distributive
algebraic way, only subject to the following rule for finding
the product of two elementary relations of shapes a:b and c:d.

| (a:b)(c:d) = (a:d) if b = c,
|
| (a:b)(c:d) = 0 otherwise.

Here 0 may be taken as the empty set {}, or anything that serves as
the "additive identity element", meaning that C + 0 = C |_| {} = C,
where C is any set.

At 07:36 PM 22/11/2002, you wrote:
>Thanks for the clarification, John. I find something baffling in the
>following, though:
>
> > As I see it, there are two issues. One is whether representation
> > and some other things involve triadic relations. The other
> > is whether there are irreducibly triadic relations. They
> > are not the same issue. So far, I find in Peirce the first
> > issue made quite convincingly in the affirmative. I have not
> > found the second case to be made convincingly at all
> > by either side.
>
>But if you are persuaded of the first, why are you not persuaded ipso facto
>of the second? Is there some recondite sense of "reducible" involved in
>this?

Not at all, Joe. It is possible for a relation to be triadic and
for it to be reducible. I gave examples in my post. There is
nothing especially difficult or tricky in what I am saying.
Quite the contrary. Between, as I mentioned, is a triadic
relation, but it can be composed of dyadic relations. For example,
if we say that the ham is between the upper and lower slices
of bread, that is equivalent to saying that it is below the upper
and above the lower. More formally Between(ham,upper,lower)
iff Above(upper, lower) and Above(ham,lower) and Above(upper,
ham). Between is clearly a triadic relation, and the location of the
ham in this case involves it, but the relation can be analyzed fully
into dyadic relations.

JC: It is possible for a relation to be triadic and
for it to be reducible. I gave examples in my
post. There is nothing especially difficult or
tricky in what I am saying. Quite the contrary.
Between, as I mentioned, is a triadic relation, but
it can be composed of dyadic relations. For example,
if we say that the ham is between the upper and lower
slices of bread, that is equivalent to saying that it
is below the upper and above the lower. More formally
Between(ham,upper,lower) iff Above(upper, lower) and
Above(ham,lower) and Above(upper, ham). Between is
clearly a triadic relation, and the location of the
ham in this case involves it, but the relation can
be analyzed fully into dyadic relations.

John Collier has provided us with a perfect example --
a perfect example of someone who does not know what
a relation is, does not know what the difference
between a relation and one of its instances is,
does not therefore know what either a 3-adic
or a 2-adic relation is, and does not know
what decomposition or reduction is.